Nonlocal Phase Transitions with Singular Heterogeneous Kernels

TL;DR

This paper investigates a family of non-local functionals related to phase transitions, generalizing previous work, and computes the limit energy as an anisotropic surface energy using $ ext{Gamma}$-convergence.

Contribution

It extends known results by analyzing singularly perturbed non-local functionals with heterogeneous kernels, including those related to fractional Sobolev spaces.

Findings

Limit energy is an anisotropic surface energy on phase interfaces.

Generalization of previous results to broader class of kernels.

Application of $ ext{Gamma}$-convergence to compute the limit energy.

Abstract

In this paper the study of a non-local Cahn-Hilliard-type singularly perturbed family of functionals is undertaken, generalizing known results by Alberti & Bellettini. The kernels considered include those leading to Gagliardo seminorms for fractional Sobolev spaces. The limit energy is computed via $\Gamma$-convergence and shown to be an anisotropic surface energy on the interface between the two phases.